### 6.2 Metric-based perturbations

An alternative approach to brane-world cosmological perturbations is an extension of the 4D
metric-based gauge-invariant theory [170, 241]. A review of this approach is given in [40, 269].
In an arbitrary gauge, and for a flat FRW background, the perturbed metric has the form
where the background metric functions are given by Equations (181, 182). The scalars
represent scalar perturbations. The vectors , , and are transverse, so that
they represent 3D vector perturbations, and the tensor is transverse traceless, representing 3D tensor
perturbations.
In the Gaussian normal gauge, the brane coordinate-position remains fixed under perturbation,

where is the background metric, Equation (180). In this gauge, we have
In the 5D longitudinal gauge, one gets

In this gauge, and for an background, the metric perturbation quantities can all be expressed in
terms of a “master variable” which obeys a wave equation [244, 246]. In the case of scalar
perturbations, we have for example
with similar expressions for the other quantities. All of the metric perturbation quantities are determined
once a solution is found for the wave equation
The junction conditions (62) relate the off-brane derivatives of metric perturbations to the matter
perturbations:

where
For scalar perturbations in the Gaussian normal gauge, this gives
where is the scalar potential for the matter anisotropic stress,
The perturbed KK energy-momentum tensor on the brane is given by
The evolution of the bulk metric perturbations is determined by the perturbed 5D field equations in the
vacuum bulk,
Then the matter perturbations on the brane enter via the perturbed junction conditions (273).
For example, for scalar perturbations in Gaussian normal gauge, we have

For tensor perturbations (in any gauge), the only nonzero components of the perturbed Einstein tensor are
In the following, I will discuss various perturbation problems, using either a 1+3-covariant or a
metric-based approach.